"Beta" or "beta reader" is a fandom term for a person or group of people who will read your story and offer suggestions for improvement before you publish it. Beta readers are volunteer proofreaders, continuity checkers, sounding boards, and general critique providers. Not every beta reader excels at every part of that job description, and you may need one type of beta more than you need another.
It's hard to find a good beta. I've heard people say that they just don't have time to have their stories beta read. I've had betas flake out on me mid-story and disappear without a trace (In fact, I have a confession to make. Back in the early 2000s, I volunteered to beta read someone's AU, my computer died, and by the time I got a new one, I had lost that person's contact info and the Yahoo group that we met on was no longer in existence — so, to the girl whose story I flaked on, I'm very sorry and I hope that one day you found a beta reader who helped you out, because you had a really good premise and I liked the chapters that I saw.) I've also had beta readers who seem to find fault with every single thing I wrote and every word choice I made. There's a fine line between being a tough editor and being an asshole, and some people just don't know where that line is.
So, given all of that, why would I say that every good fic writer needs a good beta?
It's pretty simple. I've learned that no matter how good I think my grammar is, no matter how many times I read a chapter over, no matter how thorough I think I am in covering every plot hole and every eventuality in the story, I will make a mistake. For example, in my big Star Wars AU, Luke loses Anakin Skywalker's lightsaber — you know, the one that Ben Kenobi gave him and told him that his father wanted him to have it. He loses it in a duel, and Darth Vader is able to retrieve it, so it is in Vader's possession the next time the two of them meet. I wrote that second scene out of order, before I realized that Luke was going to lose the lightsaber, and I got so caught up in what I was writing that I completely forgot about it until a reader caught it, and by that time, the goof had been online for several months. Now, I was able to fix it just by changing a few lines, but a good beta reader would've caught it before the section ever went online. I can think of several authors — and I'm not naming names, because I think that would be mean — who come up with great ideas and obviously love their characters, but write stories that desperately need a second pair of eyes, not only for grammar and punctuation but for repetitive or pedantic narrative styles that hurt their work more than they're aware.
My writing style has certainly changed since I began writing fanfiction, and I make no claim to be an expert. I don't always have a beta reader. I can't always find one, but I can say without a doubt that I can tell a clear difference in the quality of my work if I have a beta versus if I don't. I have a problem with reading what the page is supposed to say instead of what it says. (People usually suggest reading the material out loud or reading it backwards. I have tried both of those things for years and neither one of them is very effective for me, but you may have a different experience. Go ahead and try it if you can't find yourself a beta.) Even if the beta doesn't find very much or if all they're looking for is spelling, grammar, etc, they'll find something, and that will lead you to go back in and take a look at the story more closely. That is never a bad thing.
Lots of times we're just impatient to post. We get excited. We spent all this time and energy working on a chapter; we have something we think other fans will really like, and we want to share it RIGHT now. We may feel pressure because we know that we have readers waiting and asking for updates, especially if the story's been around for a while. No one ever died from waiting a week or two.
Sometimes we are uncomfortable showing our work before it's finished. Maybe we don't write in chronological order; maybe our rough drafts don't make very much sense, and by the time we put them together in a way that would make sense to another person, we figure a beta reader really isn't necessary. My experience has been that a good beta reader can help with all of these things (or in spite of them) and even if your story is fine without one, having one will make it that much better.